Subgroups — C₁₃₇×#7842 #46857228

Order 35072.

558 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1744×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₄×#97×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₄²×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#322×C₁₃₇	C₂	1	2	yes	maximal
5	C₂×#322×C₁₃₇	C₂	1	2	yes	maximal
6	C₂×#97×C₁₃₇	C₄	1	4	yes	—
7	C₂×C₄×C₈×C₁₃₇	C₄	1	4	yes	—
8	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
12	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
13	#336×C₁₃₇	C₄	1	4	yes	—
14	#322×C₁₃₇	C₄	1	8	yes	complemented
15	#322×C₁₃₇	C₄	1	8	yes	complemented
16	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
18	C₂²×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
19	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
24	#97×C₁₃₇	—	2	8	no	—
25	C₄×C₈×C₁₃₇	—	2	8	no	—
26	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
27	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
32	C₂²×C₄×C₁₃₇	C₄²	1	4	yes	—
33	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₃₇	—	2	4	no	—
37	C₂×C₈×C₁₃₇	—	2	8	no	—
38	C₂×C₈×C₁₃₇	—	2	8	no	—
39	C₂×C₈×C₁₃₇	C₄²	1	4	yes	—
40	C₂×C₈×C₁₃₇	C₄²	1	8	yes	—
41	C₂×C₈×C₁₃₇	C₄²	1	8	yes	—
42	C₂×C₈×C₁₃₇	C₄²	1	4	yes	—
43	C₂×C₈×C₁₃₇	M₁₆	1	8	yes	—
44	C₄²×C₁₃₇	—	2	4	no	—
45	C₄²×C₁₃₇	—	2	4	no	—
46	C₄²×C₁₃₇	M₁₆	1	8	yes	—
47	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
48	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
49	C₂³×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₄×C₁₃₇	—	2	4	no	—
51	C₂×C₄×C₁₃₇	—	2	4	no	—
52	C₂×C₄×C₁₃₇	#97	1	8	yes	—
53	C₂×C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
55	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
56	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
57	C₂×C₄×C₁₃₇	C₂×M₁₆	1	4	yes	—
58	C₂×C₄×C₁₃₇	C₂×M₁₆	1	4	yes	—
59	C₂×C₄×C₁₃₇	#131	1	4	yes	—
60	C₈×C₁₃₇	—	2	4	no	—
61	C₈×C₁₃₇	—	2	8	no	—
62	C₈×C₁₃₇	—	2	8	no	—
63	C₈×C₁₃₇	#97	1	8	yes	complemented
64	C₂²×C₁₃₇	C₄³	1	1	yes	characteristic
65	C₂²×C₁₃₇	C₂×#97	1	2	yes	—
66	C₂²×C₁₃₇	C₄×M₁₆	1	2	yes	—
67	C₂²×C₁₃₇	#405	1	2	yes	—
68	C₄×C₁₃₇	—	2	8	no	—
69	C₄×C₁₃₇	C₂×#97	1	2	yes	—
70	C₄×C₁₃₇	C₄×M₁₆	1	2	yes	—
71	C₄×C₁₃₇	C₄×M₁₆	1	4	yes	—
72	C₄×C₁₃₇	#405	1	4	yes	—
73	C₂×C₁₃₇	#1451	1	4	yes	—
74	C₂×C₁₃₇	C₄×#97	1	1	yes	characteristic
75	C₂×C₁₃₇	C₄×#97	1	1	yes	characteristic
76	C₂×C₁₃₇	#1726	1	1	yes	characteristic
77	#7842	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₁₃₇	#7842	1	1	yes	characteristic, direct product factor, Sylow
79	#1744	C₂×C₁₃₇	1	1	yes	characteristic
80	C₄×#97	C₂×C₁₃₇	1	1	yes	characteristic
81	C₄²×C₈	C₂×C₁₃₇	1	1	yes	characteristic
82	C₂×#322	C₂×C₁₃₇	1	2	yes	—
83	C₂×#322	C₂×C₁₃₇	1	2	yes	—
84	C₂×#97	C₄×C₁₃₇	1	4	yes	—
85	C₂×C₄×C₈	C₄×C₁₃₇	1	4	yes	—
86	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
87	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
89	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
90	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
91	#336	C₄×C₁₃₇	1	4	yes	—
92	#322	C₄×C₁₃₇	1	8	yes	complemented
93	#322	C₄×C₁₃₇	1	8	yes	complemented
94	C₂²×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
95	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
96	C₂²×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
97	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
98	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
99	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
100	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
101	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
102	#97	—	2	8	no	—
103	C₄×C₈	—	2	8	no	—
104	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
105	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
106	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
107	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
108	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
109	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
110	C₂²×C₄	C₄²×C₁₃₇	1	4	yes	—
111	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
112	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
113	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	2	8	no	—
116	C₂×C₈	—	2	8	no	—
117	C₂×C₈	C₄²×C₁₃₇	1	4	yes	—
118	C₂×C₈	C₄²×C₁₃₇	1	8	yes	—
119	C₂×C₈	C₄²×C₁₃₇	1	8	yes	—
120	C₂×C₈	C₄²×C₁₃₇	1	4	yes	—
121	C₂×C₈	M₁₆×C₁₃₇	1	8	yes	—
122	C₄²	—	2	4	no	—
123	C₄²	—	2	4	no	—
124	C₄²	M₁₆×C₁₃₇	1	8	yes	—
125	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
126	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
127	C₂³	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	#97×C₁₃₇	1	8	yes	—
131	C₂×C₄	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
132	C₂×C₄	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
133	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
134	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
135	C₂×C₄	C₂×M₁₆×C₁₃₇	1	4	yes	—
136	C₂×C₄	C₂×M₁₆×C₁₃₇	1	4	yes	—
137	C₂×C₄	#131×C₁₃₇	1	4	yes	—
138	C₈	—	2	4	no	—
139	C₈	—	2	8	no	—
140	C₈	—	2	8	no	—
141	C₈	#97×C₁₃₇	1	8	yes	complemented
142	C₂²	C₄³×C₁₃₇	1	1	yes	characteristic
143	C₂²	C₂×#97×C₁₃₇	1	2	yes	—
144	C₂²	C₄×M₁₆×C₁₃₇	1	2	yes	—
145	C₂²	#405×C₁₃₇	1	2	yes	—
146	C₄	—	2	8	no	—
147	C₄	C₂×#97×C₁₃₇	1	2	yes	—
148	C₄	C₄×M₁₆×C₁₃₇	1	2	yes	—
149	C₄	C₄×M₁₆×C₁₃₇	1	4	yes	—
150	C₄	#405×C₁₃₇	1	4	yes	—
151	C₂	#1451×C₁₃₇	1	4	yes	—
152	C₂	C₄×#97×C₁₃₇	1	1	yes	characteristic
153	C₂	C₄×#97×C₁₃₇	1	1	yes	characteristic
154	C₂	#1726×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²×C₁₃₇	C₂³
commutator	C₂²	C₄³×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
socle, prime radical	C₂³×C₁₃₇	C₂×C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
14. #322×C₁₃₇	C₄	semidirect
15. #322×C₁₃₇	C₄	semidirect
63. C₈×C₁₃₇	#97	semidirect
77. #7842	C₁₃₇	direct (N × H)
78. C₁₃₇	#7842	direct (N × H)
92. #322	C₄×C₁₃₇	semidirect
93. #322	C₄×C₁₃₇	semidirect
141. C₈	#97×C₁₃₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#1744×C₁₃₇	2
C₄×#97×C₁₃₇	2
C₄²×C₈×C₁₃₇	2
C₂×#322×C₁₃₇	2
#7842	137

Subgroups — C137×#7842 #46857228

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#7842 #46857228